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It's getting pretty late, so I'm going to come back and try to figure this out tomorrow. Thank you for all your help so far! 

Definitely!
I'll check back tomorrow.

It might be a bad frame ground, or the switch itself might be dodgy. But given the turn-signals work and the hazards don’t, it’s probably a switch issue or hazard-circuit issue. 

You could try opening up the switch and giving it a blast of contact cleaner. Check for corrosion in the switch contacts, sand lightly if needed, add a little bit of lithium grease, and reassemble. Check, clean, and grease the electrical connections, too. 

wrote:

It might be a bad frame ground, or the switch itself might be dodgy. But given the turn-signals work and the hazards don’t, it’s probably a switch issue or hazard-circuit issue. 

You could try opening up the switch and giving it a blast of contact cleaner. Check for corrosion in the switch contacts, sand lightly if needed, add a little bit of lithium grease, and reassemble. Check, clean, and grease the electrical connections, too. 

Thanks for the tip - I haven't really been working on this issue, as I was able to pass inspection without my hazards, so fixing them suddenly fell pretty low on my list of priorities. But your suggestions are definitely worth trying once I revisit this task!